perenflo.com

A history of solving

some famous problems

in mathematical analysis

This paper provides

a historical view of

what functional analysis is,

a biography of Per H. Enflo

and a short view of four

fundamental problems in the

geometry of Banach spaces.

It gives the story of how Enflo, after several years of work, managed to solve three of these problems. It is a story of trying many ideas, testing them on simpler problems, trying to combine different techniques for handling different aspects of the problems, making many failed attempts and finally succeeding.

In 1966 Enflo found a general strategy for solving many problems in infinite dimensions, by combining appropriate finite-dimensional techniques with induction procedures. When applied to the old, fundamental problems in functional analysis, this has led to new concepts, problems, techniques and results in analysis. These developments have been applied also in several other areas of mathematics and computer science.

Nordic Mini-Conference in Algebra, Analysis, and Geometry MDU November 6-8, 2024

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VOTCAM 2024 - Virginia Operator Theory and Complex Analysis Meeting

on October 26

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Visiting Speaker to the UK - June 3 to June 14 2024

Valencia, Departamento de Análisis Matemático, Universidad de Valencia

WORKSHOP ON FUNCTIONAL ANALYSIS, November 16th 2023 >> YouTube >>

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Copenhagen, Department of Mathematical Sciences, University of Copenhagen 2024

Special Analysis Lecture Wednesday, April 17>>

Piano Recital, Thursday, April>>

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Lund, Centre for Mathematical Sciences, Lund University

Colloquium, September 13th 2023 >>

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Helsinki, IWOTA 2023, University of Helsinki >>

July 31 - August 4, 34th International Workshop on Operator Theory and its Applications,

Semi-plenary session Tuesday August 1st
Concert August 3rd Conference Dinner -

Beethoven, Waldstein

Schubert, Impromptu in G flat major

Schumann, Vogel als Prophet

Per Enflo (b. 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved several fundamental problems. Three of these problems had been open for more than forty years: The Basis Problem, The Approximation Problem and The Invariant Subspace Problem for Banach Spaces.
In solving these problems, Enflo developed new concepts and techniques, which were then used by other researchers in several mathematical fields, in algebra, number theory, classical and functional analysis, operator theory and in computer science, especially computer algebra and approximation algorithms. (see Wikipedia)

Per Enflo is also a renowned concert pianist, who has been concertizing for almost 70 years on five continents. Numerous recent recordings as soloist, soloist with orchestras, chamber musician and accompanist may be mentioned as well as a critic-acclaimed duo recital in Carnegie Hall, NY, with violinist Hristo Popov.

Enflo worked at Kent State University [1989-2012] where he held the title of University Professor. Enflo held earlier positions at the Miller Institute for Basic Research in Science at the University of California, Berkeley, Stanford University, Ecole Polytechnique, (Paris) and The Royal Institute of Technology, Stockholm.

Upcoming Symposium June 16-20 2025, Complutense University of Madrid >>

Årsmöte maj 2025 Svenska Matematikersamfundet

Distinguished Lecture: Construction of Invariant Sub-

spaces for Operators on Hilbert Space

<< Mathematics >> << Music >> << Interviews & Reviews >>

Per Enflo - mathematician and pianist -

was the last laureate of a prize for solving

scientific problem from the Scottish Book.

ON THE INVARIANT SUBSPACE PROBLEM

IN HILBERT SPACES

PER H. ENFLO

Abstract. In this paper we show that every bounded linear operator T on a Hilbert

space H has a closed non-trivial invariant subspace

Online: click here >> arXiv:2305.15442 [pdf,ps,other]

[Submitted on 24 May 2023 (v1), last revised 6 Apr 2024 (this version, v2)]

Proceedings of Symposia in Pure Mathematics >>

Volume:106;2024; 346 pp

MSC:Primary 43; 46; 47

This book provides a comprehensive presentation of recent approaches to and results about properties of various classes of functional spaces, such as Banach spaces, uniformly convex spaces, function spaces, and Banach algebras. Each of the 12 articles in this book gives a broad overview of current subjects and presents open problems. Each article includes an extensive bibliography.

This book is dedicated to Professor Per. H. Enflo, who made significant contributions to functional analysis and operator theory.

A Few Lines on Per Enflo’s Works
Gilles Godefroy, Mohammad Sal Moslehian, and Juan Benigno Seoane-Sepúlveda
Abstract. A brief historical overview of the notable mathematical contributions of Per Enflo is presented.

Readership

Graduate students and researchers interested in functional analysis and operator theory.

IBM original poster from the 1970:s >>>

P H ENFLO
Department of Mathematical Sciences, Kent State University, Kent, OH, USA and
Instituto de Matemática Interdisciplinar, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Spain

M S MOSLEHIAN
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran
J B SEOANE-SEPÚLVEDA
Instituto de Matemática Interdisciplinar (IMI), Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid, Spain

Published: June 11, 2023

Invariant Subspace Problem in Hilbert Spaces:

Exploring Applications in Quantum Mechanics, Control Theory, Operator Algebras, Functional Analysis and Accelerator Physics - Mostafa Behtouei arXiv:2306.17023

"Per H. Enflo’s recent claim of solving the invariant subspace problem in his recent publication titled ”On the invariant subspace problem in Hilbert spaces.” [4] has generated excitement and anticipation in the mathematical community, as his work could represent a major breakthrough in the field. Enflo’s solution, if validated,would be a groundbreaking achievement in mathematics, resolving a longstanding problem and advancing our understanding of bounded linear operators on Hilbert spaces. Its impact would be significant and far-reaching." From: Mostafa Behtouei, 29 Jun 2023

When did you make the decision to go into math instead of music?

I have never made, and will never make, any such decision

Craiova, Romania, the 20 th October 2023

"Invited to participate as a soloist in the Concert no. 1 in C major for piano and orchestra, op. 15 of Ludwig van Beethvoven was Per Enflo.

A Swede settled in the United States, this respectable gentleman is not only a pianist so good that he has touched all the meridians of the world with his art, but also a renowned mathematician.

Yes, mathematician.

Two fields that seem so far apart, but which, at the highest level, approach asymptotically. Let us think only of Pythagoras and the sacred importance he attached to music.

Per Enflo has not only a remarkable destiny, but also a long-lasting one, which I wish twice as long.

At 80, he still has the energy to let himself be swept up in the gaiety of Beethoven's composition with the enthusiasm of a toddler (until his piece began, he seemed to be struggling to resist the temptation to touch the keyboard).

What is the secret of such a zest for life at such an advanced age?

But perhaps that is not the question, but the answer.

With his lordship's permission, I will also draw a parallel between the way he crossed his hands on the piano keys and the way of multiplying the means and extremes of a proportion.

Photos by my friend Daniel Botea, who, although he is incredibly old, enjoyed the same as me, like a child that is, all the wonders that the concert offered.

Autor: Iris Gottlieb (sursa – www.vox.com)

Google translation from original Romanian, Facebook:

https://www.iulianfira.ro/category/intre-muzical-si-conceptual/

"Mathematics and Music - What is the connection?"
Lecture >> 31 August 2021, Science Festival Strömstad Academy Slides >
Beethoven last piano sonata Concert >> Strömstad Academy's Scientific Festival 2021

Mozart - Beethoven - Chopin Concert >> Jubilee Congress of Polish Mathematicians on the 100th Anniversary of Polish Mathematical Society (2019, Krakow)

"An infinite-dimensional version of Hilbert's fifth problem with connections to non-linear geometry of Banach spaces"
Lecture December 2021 Slides >> to Holiday Lecture by Per Enflo via Zoom - Jani Virtanen (Reading and Helsinki) & Håkan Hedenmalm (KTH and Reading)